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MTH301 - Calculus II GDB No. 01 Solution and Discussion Fall 2013 Due Date:20th February, 2014

GDB Topic Dated: Feb 14, 14

GDB Topic:
Discuss the applications of  partial derivatives in daily life with at least 2 examples.

Opening Date: Wednesday, 19th February, 2014 at 12:01 AM

Closing Date: Thursday, 20th February, 2014 at 11:59 PM

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(1) Maxwell's equations of electromagnetism
(2) Einstein's general relativity equation for the curvature of space-time given mass-energy-momentum.
(3) The equation for heat conduction (Fourier)
(4) The equation for the gravitational potential of a blob of mass (Newton-Laplace)
(5) The equations of motion of a fluid (gas or liquid) (Euler-Navier-Stokes)
(6) The Schrodinger equation of quantum mechanics
(7) The Dirac equation of quantum mechanics
(8) The Yang-Mills equation
(9) The Liouville equation of statistical mechanics

A derivative of a function of one variable expresses a change in that function relative to its argument. For example Newtons law for rectilinear motion is F = mass times the second derivative of distance with respect to time.

But the world involves functions that depend on several variables. For example the pressure of a gas depends on density and temperature. The speed of sound (squared), it turns out, in a nebula in space (which is very nearly at constant temperature due to radiative transport) is the partial derivative of the pressure with respect to density keeping temperature fixed.

Your happiness H depends on how much money, m, you make and the number of hours, h, you spend with your family. H = H(m, h). But how much money you make also depends on how much on how much time you spend with your family. The more time you spend with them the less money you will make. So m = m(h) and we must write

H = H(m(h), h)

Now, we want to know how many hours "h" to work to maximize happiness so we take the _total_ derivative of H with respect to h and set it equal to zero:

dH/dh = (partial H/partial m)*(partial m/partial h) + (partial H/partial h) = 0.

The famous partial differential equation is the wave or Harmonic equation in physics. Then there is the Navier-Stokes equation in fluid mechanics. Just about any situation with multiple variables can be modeled with some partial differential equations. The problem is not so much as how the partial derivatives arise, but how to solve it. 

What you learn in a partial differential equation class is not how it is actually solved in real-life. They only give you the simple cases. In real-life engineering, you have software programs using numerical techniques like finite differences to solve the partial derivative problems. 

For example, the space shuttle foam problem during lift-off can be modeled with the Navier-Stokes partial differential equation. I believe it has 7 variables. If you leave all 7 variables unconstrained fluid flow over the shuttle structure, then it will probably require a few parallel supercomputing servers to crunch the numbers to look at all the different possibilities (or solutions) of how a foam may fall during lift-off. The results give the engineers an idea of what to expect with a free flowing foam near the Shuttle during lift-off and how they can design things differently to avoid any free flowing objects.

A derivative of a function of one variable expresses a change in that function relative to its argument. For example Newtons law for rectilinear motion is F = mass times the second derivative of distance with respect to time. 

But the world involves functions that depend on several variables. For example the pressure of a gas depends on density and temperature. The speed of sound (squared), it turns out, in a nebula in space (which is very nearly at constant temperature due to radiative transport) is the partial derivative of the pressure with respect to density keeping temperature fixed.

Your happiness H depends on how much money, m, you make and the number of hours, h, you spend with your family. H = H(m, h). But how much money you make also depends on how much on how much time you spend with your family. The more time you spend with them the less money you will make. So m = m(h) and we must write 

H = H(m(h), h)

Now, we want to know how many hours "h" to work to maximize happiness so we take the _total_ derivative of H with respect to h and set it equal to zero:

dH/dh = (partial H/partial m)*(partial m/partial h) + (partial H/partial h) = 0. 


Partial derivatives are used in solving sets of nonlinear equations and in min/max optimization analysis (i.e. set partial derivatives equal to zero to find critical points). 

partial differential equations abound in all branches of science and engineering and many areas of business. The number of applications is endless.

The famous partial differential equation is the wave or Harmonic equation in physics. Then there is the Navier-Stokes equation in fluid mechanics. Just about any situation with multiple variables can be modeled with some partial differential equations. The problem is not so much as how the partial derivatives arise, but how to solve it.

What you learn in a partial differential equation class is not how it is actually solved in real-life. They only give you the simple cases. In real-life engineering, you have software programs using numerical techniques like finite differences to solve the partial derivative problems.

For example, the space shuttle foam problem during lift-off can be modeled with the Navier-Stokes partial differential equation. I believe it has 7 variables. If you leave all 7 variables unconstrained fluid flow over the shuttle structure, then it will probably require a few parallel supercomputing servers to crunch the numbers to look at all the different possibilities (or solutions) of how a foam may fall during lift-off. The results give the engineers an idea of what to expect with a free flowing foam near the Shuttle during lift-off and how they can design things differently to avoid any free flowing objects.

The famous partial differential equation is the wave or Harmonic equation in physics. Then there is the Navier-Stokes equation in fluid mechanics. Just about any situation with multiple variables can be modeled with some partial differential equations. The problem is not so much as how the partial derivatives arise, but how to solve it.

    What you learn in a partial differential equation class is not how it is actually solved in real-life. They only give you the simple cases. In real-life engineering, you have software programs using numerical techniques like finite differences to solve the partial derivative problems.

    For example, the space shuttle foam problem during lift-off can be modeled with the Navier-Stokes partial differential equation. I believe it has 7 variables. If you leave all 7 variables unconstrained fluid flow over the shuttle structure, then it will probably require a few parallel supercomputing servers to crunch the numbers to look at all the different possibilities (or solutions) of how a foam may fall during lift-off. The results give the engineers an idea of what to expect with a free flowing foam near the Shuttle during lift-off and how they can design things differently to avoid any free flowing objects.

Applications of Partial Derivatives
In this section we will take a look at a couple of applications of partial derivatives. Most of the applications will be extensions to applications to ordinary derivatives that we saw back in Calculus I. For instance, we will be looking at finding the absolute and relative extrema of a function and we will also be looking at optimization. Both (all three?) of these subjects were major applications back in Calculus I. They will, however, be a little more work here because we now have more than one variable.

Here is a list of the topics in this chapter.

Tangent Planes and Linear Approximations We’ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.

Gradient Vector, Tangent Planes and Normal Lines In this section we’ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.

Relative Minimums and Maximums Here we will see how to identify relative minimums and maximums.

Absolute Minimums and Maximums We will find absolute minimums and maximums of a function over a given region.

Lagrange Multipliers In this section we’ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.

Partial derivatives are used in solving sets of nonlinear equations and in min/max optimization analysis (i.e. set partial derivatives equal to zero to find critical points). 

partial differential equations abound in all branches of science and engineering and many areas of business. The number of applications is endless.

A derivative of a function of one variable expresses a change in that function relative to its argument. For example Newtons law for rectilinear motion is F = mass times the second derivative of distance with respect to time. 

But the world involves functions that depend on several variables. For example the pressure of a gas depends on density and temperature. The speed of sound (squared), it turns out, in a nebula in space (which is very nearly at constant temperature due to radiative transport) is the partial derivative of the pressure with respect to density keeping temperature fixed.

Your happiness H depends on how much money, m, you make and the number of hours, h, you spend with your family. H = H(m, h). But how much money you make also depends on how much on how much time you spend with your family. The more time you spend with them the less money you will make. So m = m(h) and we must write 

H = H(m(h), h)

Now, we want to know how many hours "h" to work to maximize happiness so we take the _total_ derivative of H with respect to h and set it equal to zero:

dH/dh = (partial H/partial m)*(partial m/partial h) + (partial H/partial h) = 0. 

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